Math Problem!

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Problem...

Two positive real numbers, A and B, are selected at random.

What is the probability that A squared is less than B?

^^ I must say, I am rather confused by this question. I believe that this equivalent to calculating the probability that A is less than the positive square root of B. ^^ However, this is just calculating the probability that one random positive real number is less than another random positive real number. ^^ I am very sorry if this is incorrect of me, however I would suspect that the answer is zero. ^^ I believe this as if you consider a fixed number root B, then are an infinite number of numbers greater than root B (let us call this C from now on). However, there are only C numbers less than C (including zero), which is finite. So the numbers that are greater than C greatly outnumber the number of numbers less than C but greater than zero (by an infinite amount). Thus I would guess that the answer is zero.

EDIT: ^^ Oh dear please wait, I believe this is incorrect, as I am considering one fixed number. I shall now write a revised message and have placed this here to notify of my blunder.

EDIT 2: ^^ Righty, I do believe my answer above is incorrect, yet I am uncertain of the correct answer. ^^ I believe it is best to consider invdividual cases of fixing C. If you fix C very close to zero, then the probability that A is less than C is zero. However, if we consider as C tends to infinity, then the probability that A is less C also tends to infinity. ^^ I hope this reasoning assists you. ^^ I am sorry if I am very incorrect and being silly.

Last edited by Tomasu on 12 Aug 2009, 2:15 pm, edited 1 time in total.

I'd guess 25%

50% chance B is larger than A
50% chance A is larger than B

Considering you're dealing with infinite numbers, I'd say A squared has just as good of a chance of being greater than B as it does less. Cutting 50% in half, you get 25%.

An infinitely small but non-zero probability, or what probability theory calls a de minimis probability.

Tomasu has the right idea, but got confused with his inclusion of "C". Deadeyexx assumed that the probabilities of A^2 > B and A^2 < B are equivalent. They are, based on the ideas presented by Tomasu, not equivalent. In other words, this question cannot be treated as a coin-tossing problem, with the probability of two heads coming up when two fair coins are tossed being the desired answer.

Infinite sets create infinite confusion in math. Selecting a member of an infinite set is always a de minimis probability, non-zero but infinitely small. That's where the confusion in this question arises.

Note: This is my analysis, and if I'm incorrect, please advise. Probability theory is something that fascinates me, but I am not a master of it.

^^ Yaye I believe I rather understand. ^^ Perhaps if you have studied continuous random variables, I believe this may certainly assist with your reasoning. ^^ I am a little rusty with Probability Theory as of the moment as I am being blessed by a happy holiday and feel very evil and am therefore very sorry. ^^ I believe that the probability of a continuous random variable being equal to fixed point (that is, one number) is equal to zero, which I believe is what happy Aoi has stated. ^^ Thankees Aoi and PetitPrince and deadeyexx, I do enjoy Mathematics problems.

^^ Thinking of this however, I remain rather uncertain, as we are not considering the probability of choosing a fixed number, we are considering the prbability of A^2 being an element of the interval [0,B], however A and B are both not fixed. ^^ To attempt to help with this, I tried to avoid this problem by fixing B, and the seeing what happens as B is very small and as it then tends to infinity. ^^ Sorry if I have been silly.

I think I get it now. The chances of B being within the bounds of A & A^2 are infinately smaller than the chances of B being in the A^2 to infinity bracket.

If I may respectfully disagree, collegue Aoi. De minimis would be true if someone were to ask "What is the probability that A is 17" for example. Since set A is infinite, the answer is de minimis. However, this is not quite the question in this case. As further proof to make this obvious, I offer the following randomly selected A and B:

A=2.1242
B=17.562 A squared is less than B

A=0
B=6 A squared is less than B

With these two examples, A squared is indeed less than B which makes a de minimis assesment even more unlikely and "zero" would be impossible (apologies to friend Tomasu).

Infinity is actually your ally here and is the key to simplification of the answer. I provide the following hint:

Both A and B come from infinite sets [0,inf] in |R| (positive reals). This renders the mathematical operation of squaring A completely irrelevant. If B is 17 (keeping it whole for the sake of simplicity), how many numbers are LESS than the square root of 17? There are an infinite number of positive reals less than the square root of 17. Therefore simplify this down to "What is the probability A is less than B". It's the same answer.

To further assist, I will give the three resultant possibilites. A is less than B, A is greater than B, and A = B. Does this make it easier to see? What is the answer?

Thank you for the correction. I agree with your analysis. I made an error when considering the real numbers.

As you point out, between any two given real numbers there is an infinity of numbers.

No worries! One of the most beautiful things about the people on this site is their ability to simply change their mind without emotion or prejudice when presented with contrary evidence. I think it is a very admirable quality.

Also, that was a great analysis of the probability of A being any particular number. Very logical and well-reasoned stemming from dice/cards. You've either done some study of this or you are naturally very logical (or both )

^^ I am very sorry if I have not been of help, I hang my head in shame. ^^ I believe I do understand now Vipera Aspis. ^^ However, if you read my previous posts again, I believe I did correct myself concerning the answer being zero, and gave an analysis similar to your own, Vipera Aspis. In my second post in this thread, I did initially agree with Aoi, yet then changed my mind and thought again. I am very sorry if I have been horrible. ^^ I do believe I am very incorrect, and you certainly did very well Vipera Aspis and Aoi, and I was incorrect, I am sorry.

Oh no, Tomasu! Not at all! You have a beautiful mind. ^^I am always happy to read your posts! They cheer me up with happy words and you always try to help others. Don't be sad. You thought for a long time and thought hard. You helped others think hard. And it's just a silly math problem, it is okay if you don't get "the answer". Like life, it is about the journey and not the solution! I'm happy you took a shot at it! You are allowed to swing and miss. It is totally okay. Everybody does it! You can't improve without doing it!

Come back happy Ametomasu! Come back! You are really liked here!

^^

The answer is zero.

Good job, Tomasu.

^^ Yaye thankees Vipera Aspis, I am sorry for confusion. ^^ Wowwees, I see LePetitPrince, however, I believed my first answer, zero, to be incorrect. Was B perhaps fixed may I ask? ^^ Yaye thankees Vipera Aspis, I am very sorry if this seemed that I had departed. ^^ I believe you do all possess very happy and beautiful minds here, thankees everyone. ^^ I did enjoy this problem.

Proof requested

Is it a trick question, like there is non-uniform distribution? 'Cause there seem to be an infinite number of cases that go against the answer (given the current info).

Example:
A = 5
B = 100.141
A ^ 2 = 25
25 < 100.141 therefore A^2 is less than B which should invalidate a 'zero' probability of the occurence.

Plus those values came from a program I wrote that actually DOES pick two numbers randomly from positive reals, squares one and returns the comparison. So...

This may be a philosophy question, not a math question, but isn't the use of infinity render the issue somewhat undefined?

I mean, take ANY number for A. Square it. Now, there are a finite number of integers between 0 and A^. However, there are an infinite number of integers above A^. So the odds of B being below A^ would approach zero and the odds of B being above A^ would approach infinite.

Well, it's been decades since I did any math or philosophy, so I'm probably missing something. It just seems to me that when you refer to infinity in mathematics, it renders the problem fundamentally undefined because infinity is not a number, but an expectation. An expectation that something can be both limitless yet defineable. Anyway, in posing that one would pick any positive integers, that could span from 1 to infinity minus 1.

I guess the answer differs depending upon the order of picking the numbers. I mean, if you pick a number for B, then clearly there are a finite number of positive integers below that and infinite initegers above that, so the odds approach zero that, if A is a postiive integer, it will be less than B -- and this is true even without squaring A.

In the end, I am left thinking if you had to do simultaneous random selection, the answer would approach, but never reach, zero.

Wait...isn't this a case where you end up with a "smaller' infinite set over a "larger" infinite set, but because both sets are infinite, the concepts of larger and smaller become meaningless, so it is really 50/50?

I give up.

This problem statement is incomplete.

You have not defined what you mean by "selected at random".

You could restate the problem as "are selected at random, from a uniform distribution in the interval zero to K". In this case, there is a simple answer to the question, which I'll leave it to you to evaluate.

You could then allow K to tend towards large numbers, but you should not attach too much meaningfulness to the resultant answer (that A squared is infinitely likely to be larger than B), as there is not really a probability distribution that is uniform across the range zero to infinity.

There are, however, an infinite number of non-uniform probability distributions that are sensibly defined (and everywhere non-zero) on the range zero to infinity. You could choose one of them. A Poisson distribution might (or might not) be a good choice.